ハルトークスの拡張定理のソースを表示

数学の、特に[[多変数複素函数論]]において、'''ハルトークスの拡張定理'''(Hartogs' extension theorem)とは、多変数[[正則函数]]の[[特異点 (数学)|特異点]]に関する定理である。
この定理は、多変数正則関数の特異点の[[関数の台|台]]が[[コンパクト空間|コンパクト]]にならないこと、つまりおおざっぱに言うと、特異点集合がある方向に「無限遠まで伸びる」ということを述べている。
より正確には、この定理は {{Math|''n''>1}} 個の複素変数をもつ[[解析函数]]に対して、その[[孤立特異点]]がつねに[[可除特異点|除去可能特異点]]であることを示している。
この定理の最初のバージョンは、[[フリードリヒ・ハルトークス]]により証明され<ref name="hartogs">原論文である{{Harvtxt|Hartogs|1906}} や{{harvtxt|Osgood|1963|pp=56–59}}, {{harvtxt|Severi|1958|pp=111–115}}, {{harvtxt|Struppa|1988|pp=132–134}} による様々な歴史的研究報告を参照。特に最後の参考文献の p. 132 では、筆者が「{{harv|Hartogs|1906}} のタイトルで触れられており、すぐに分かる通り、証明のためのキーとなるツールは[[コーシーの積分公式]]である。」と直接言及している。</ref>、「ハルトークスの補題」や「ハルトークスの原理」としても知られている。初期の[[ソ連]]の文献では、<ref>たとえば、{{harvtxt|Vladimirov|1966|p=153}} を参照。この文献では、読者に証明のために書籍 {{harvtxt|Fuks|1963|p=284}} を紹介している。（しかし、前者の文献では、p 324 の証明は正しくない。）</ref> この定理は'''オズグッド・ブラウンの定理'''(Osgood-Brown theorem)とも呼ばれ、後の{{仮リンク|ウィリアム・フォッグ・オズグッド|en|William Fogg Osgood}}(William Fogg Osgood)と{{仮リンク|アーサー・バートン・ブラウン|en|Arthur Barton Brown}}(Arthur Barton Brown)の仕事としても知られている<ref>See {{harvtxt|Brown|1936}} and {{harvtxt|Osgood|1929}}.</ref>。この多変数の正則函数の性質は'''[[#ハルトークス現象|ハルトークス現象]]'''(Hartogs' phenomenon)とも呼ばれている。しかし、「ハルトークス現象」という表現は、[[偏微分方程式]]系や[[畳み込み|畳み込み作用素]]の解がハルトークス形式の定理を満たすという性質を表すことにも同様に使われる<ref>See {{harvtxt|Fichera|1983}} and {{harvtxt|Bratti|1986a}} {{harv|Bratti|1986b}}.</ref>。
<!--In mathematics, precisely in the theory of functions of [[several complex variables]], '''Hartogs' extension theorem''' is a statement about the [[Singularity (mathematics)|singularities]] of [[holomorphic function]]s of several variables. Informally, it states that the [[Support (mathematics)|support]] of the singularities of such functions cannot be [[compact space|compact]], therefore the singular set of a function of several complex variables must (loosely speaking) 'go off to infinity' in some direction. More precisely, it shows that the concept of [[isolated singularity]] and [[removable singularity]] coincide for [[analytic function]]s of {{math|''n'' > 1}} complex variables. A first version of this theorem was proved by [[Friedrich Hartogs]],<ref name="hartogs">See the original paper of {{Harvtxt|Hartogs|1906}} and its description in various historical surveys by  {{harvtxt|Osgood|1963|pp=56–59}}, {{harvtxt|Severi|1958|pp=111–115}} and {{harvtxt|Struppa|1988|pp=132–134}}. In particular, in this last reference on p. 132, the Author explicitly writes :-"''As it is pointed out in the title of {{harv|Hartogs|1906}}, and as the reader shall soon see, the key tool in the proof is the [[Cauchy integral formula]]''".</ref> and as such it is known also as '''Hartogs' lemma''' and '''Hartogs' principle''': in earlier [[Soviet Union|Soviet]] literature,<ref>See for example {{harvtxt|Vladimirov|1966|p=153}}, which refers the reader to the book of {{harvtxt|Fuks|1963|p=284}} for a proof (however, in the former reference it is incorrectly stated that the proof is on page 324).</ref> it is also called '''Osgood-Brown theorem''', acknowledging later work by [[Arthur Barton Brown]] and [[William Fogg Osgood]].<ref>See {{harvtxt|Brown|1936}} and {{harvtxt|Osgood|1929}}.</ref> This property of holomorphic functions of several variables is also called '''Hartogs' phenomenon''': however, the locution "Hartogs' phenomenon" is also used to identify the property of solutions of [[System of equations|systems]] of [[partial differential equation|partial differential]] or [[convolution operator|convolution equation]]s satisfying Hartogs type theorems.<ref>See {{harvtxt|Fichera|1983}} and {{harvtxt|Bratti|1986a}} {{harv|Bratti|1986b}}.</ref>-->

==歴史的な話題==
元々の証明は1906年に[[フリードリヒ・ハルトークス]]により与えられ、[[コーシーの積分公式]]を[[多変数複素函数論|多変数複素函数]]に適用して証明された<ref name="hartogs"/>。現在は、通常、{{仮リンク|ボホナー・マルティエリ・コッペルマンの公式|en|Bochner–Martinelli–Koppelman formula}}(Bochner–Martinelli–Koppelman formula)か、コンパクトな台を持つ非同次[[コーシー・リーマンの方程式]]の解に依拠して証明される。コーシー・リーマンの方程式によるアプローチは、{{仮リンク|レオン・エーレンプライス|en|Leon Ehrenpreis}}(Leon Ehrenpreis)が論文 {{Harv|Ehrenpreis|1961}} で導入した。もうひとつの非常に単純な証明は、{{仮リンク|ガエターノ・フィチェーラ|en|Gaetano Fichera}}(Gaetano Fichera)が論文 {{Harv|Fichera|1957}} で、多変数[[正則函数]]の[[ディリクレ問題]]の解と[[CR多様体#CR関数|CR関数]]に関連した概念を用いて与えた<ref>フィチェーラの証明や彼の画期的な論文 {{Harv|Fichera|1957}} は、[[多変数複素函数論]]の専門家の多くから見過ごされてきたようである。{{Harvtxt|Range|2002}} では、この分野の多くの重要な定理の正しい役割が記載されている。</ref>。後に、彼はこの定理を論文 {{Harv|Fichera|1983}} で[[偏微分方程式]]のあるクラスへ拡張し、さらにこのアイデアは、その後ギウリアーノ・バラッティ(Giuliano Bratti)により大きく拡張された<ref>See {{Harvtxt|Bratti|1986a}} {{Harv|Bratti|1986b}}.</ref>。また、金子晃らの[[微分作用素|偏微分作用素]]の日本での研究も、この分野に大きく寄与している<ref>{{Harv|Kaneko|1973}} や、そこにある文献を参照。</ref>。彼らのアプローチは、{{仮リンク|エーレンプライスの基本原理|en|Ehrenpreis' fundamental principle}}(Ehrenpreis' fundamental principle)を使うものである。
<!--==Historical note==
The original proof was given by [[Friedrich Hartogs]] in 1906, using [[Cauchy's integral formula]] for functions of [[several complex variables]].<ref name="hartogs"/> Today, usual proofs rely on either [[Bochner–Martinelli–Koppelman formula]] or the solution of the inhomogeneous [[Cauchy–Riemann equations]] with compact support. The latter approach is due to [[Leon Ehrenpreis]] who initiated it in the paper {{Harv|Ehrenpreis|1961}}. Yet another very simple proof of this result was given by [[Gaetano Fichera]] in the paper {{Harv|Fichera|1957}}, by using his solution of the [[Dirichlet problem]] for [[holomorphic function]]s of several variables and the related concept of [[CR-function]]:<ref>Fichera's prof as well as his epoch making paper {{Harv|Fichera|1957}} seem to have been overlooked by many specialists of the [[Several complex variables|theory of functions of several complex variables]]: see {{Harvtxt|Range|2002}} for the correct attribution of many important theorems in this field.</ref> later he extended the theorem to a certain class of [[partial differential operator]]s in the paper {{Harv|Fichera|1983}}, and his ideas were later further explored by Giuliano Bratti.<ref>See {{Harvtxt|Bratti|1986a}} {{Harv|Bratti|1986b}}.</ref> Also the [[Japan|Japanese school]] of the theory of [[partial differential operator]]s worked much on this topic, with notable contributions by Akira Kaneko.<ref>See his paper {{Harv|Kaneko|1973}} and the references therein.</ref> Their approach is to use [[Ehrenpreis' fundamental principle]].-->

==ハルトークス現象==
一変数で成立するが多変数では成り立たない現象を'''ハルトークス現象'''(Hartogs' phenomenon)という。この現象は、このハルトークスの拡張定理や[[正則領域]]の考え方、ひいては[[多変数複素函数|多変数複素函数論]]の発展を導いた。

2変数の場合を例にとり、<math>0 <\varepsilon < 1</math> として、二重円板 <math>\Delta^2=\{z\in\mathbb{Z};|z_1|<1,|z_2|<1\}</math> の内部領域

:<math>H_\varepsilon = \{z=(z_1,z_2)\in\Delta^2:|z_1|<\varepsilon\ \ \text{or}\ \ 1-\varepsilon< |z_2|\}</math>

を考える。

'''定理''' {{harvtxt|Hartogs|1906}}: <math>H_\varepsilon</math> 上の任意の正則函数 <math>f</math> は <math>\Delta^2</math> へ解析接続される。すなわち、<math>\Delta^2</math> 上の正則函数 <math>F</math> が存在し、<math>H_\varepsilon</math> 上で <math>F=f</math> となる。

実際、[[コーシーの積分公式]]を使い、拡張された函数 <math>F</math> 得ることができる。すべての正則函数は多重円板へ解析接続できて、多重円板はもとの正則函数が定義された領域よりも真に広くなる。このような現象は、一変数では決して起きない現象である。
<!--==Hartogs' phenomenon==
A phenomenon that holds in one variable but does not hold in several variables is called '''Hartogs' phenomenon''', which lead to the notion of this Hartorgs' extension theorem and the [[domain of holomorphy]], hence the [[Several complex variables|theory of several complex variables]].

For example in two variables, consider the interior domain

:<math>H_\varepsilon = \{z=(z_1,z_2)\in\Delta^2:|z_1|<\varepsilon\ \ \text{or}\ \ 1-\varepsilon< |z_2|\}</math>

in the two-dimensional polydisk <math>\Delta^2=\{z\in\mathbb{Z};|z_1|<1,|z_2|<1\}</math> where <math>0 <\varepsilon < 1</math> .

'''Theorem''' {{harvtxt|Hartogs|1906}}: any holomorphic functions <math>f</math> on <math>H_\varepsilon</math> are analytically continued to <math>\Delta^2</math> . Namely, there is a holomorphic function <math>F</math> on <math>\Delta^2</math> such that <math>F=f</math> on  <math>H_\varepsilon</math> .

In fact, using the [[Cauchy integral formula]] we obtain the extended function <math>F</math> , All holomorphic functions are analytically continued to the polydisk, which is restrictly larger than the domain on which the original holomorphic function is defined. Such phenomenon never happen in the case of one variable.-->
:
<!--==Formal statement==
:Let {{mvar|f}} be a [[holomorphic function]] on a [[Set (mathematics)|set]] {{math|''G\K''}}, where {{mvar|G}} is an open subset of {{math|'''C'''<sup>''n''</sup>}} ({{math|''n'' ≥ 2}}) and {{mvar|K}} is a compact subset of {{mvar|G}}. If the [[Complement (set theory)|relative complement]] {{math|''G\K''}} is connected, then {{mvar|f}} can be extended to a unique holomorphic function on {{mvar|G}}.-->

==次元 1 のときの反例==
このハルトークスの拡張定理は {{math|n {{=}} 1}} のときには成り立たない。次元 1 でこの定理が成り立たないことを示すには、函数 {{math|f(z) {{=}} z<sup>−1</sup>}} を考えれば充分である。この函数は明らかに {{math|'''C'''\{0}}} の中では正則であるが、{{math|'''C'''}} 全体上の正則函数として連続ではない。このように一変数と多変数の函数論の間の差異が顕わになることこそ、ハルトークス現象の性質である。
<!--==Counterexamples in dimension one==
The theorem does not hold when {{math|''n'' {{=}} 1}}. To see this, it suffices to consider the function {{math|''f''(''z'') {{=}} ''z''<sup>−1</sup>}}, which is clearly holomorphic in {{math|'''C'''\{0},}} but cannot be continued as an holomorphic function on the whole {{math|'''C'''}}. Therefore the Hartogs' phenomenon constitutes one elementary phenomenon that emphasizes the difference between the theory of functions of one and several complex variables.-->

== 脚注 ==
{{reflist|30em}}

==歴史的な参考文献==
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==参考文献==
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 |last        = Bratti
 |first       = Giuliano
 |title       = A proposito di un esempio di Fichera relativo al fenomeno di Hartogs
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 |series      = serie 5,
 |volume      = X
 |issue       = 1
 |pages       = 241–246
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 |language    = Italian. English [[Abstract (summary)|summary]]
 |url         = http://www.accademiaxl.it/Biblioteca/Pubblicazioni/browser.php?VoceID=2020
 |doi         = 
 |mr          = 0879111
 |zbl         = 0646.35007
 |archiveurl  = https://web.archive.org/web/20110726235834/http://www.accademiaxl.it/Biblioteca/Pubblicazioni/browser.php?VoceID=2020
 |archivedate = 2011年7月26日
 |deadurldate = 2017年9月
}}. A translation of the title reads as:-"''About an example of Fichera concerning Hartogs' phenomenon''".
*{{Citation
 |last        = Bratti
 |first       = Giuliano
 |title       = Estensione di un teorema di Fichera relativo al fenomeno di Hartogs per sistemi differenziali a coefficenti costanti
 |journal     = Rendiconti della Accademia Nazionale delle Scienze Detta dei XL
 |series      = serie 5
 |volume      = X
 |issue       = 1
 |pages       = 255–259
 |year        = 1986b
 |language    = Italian. English summary
 |url         = http://www.accademiaxl.it/Biblioteca/Pubblicazioni/browser.php?VoceID=2023
 |doi         = 
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 |archivedate = 2011年7月26日
 |deadurldate = 2017年9月
}}. An English translation of the title reads as:-"''Extension of a theorem of Fichera for systems of P.D.E. with constant coefficients, concerning Hartogs' phenomenon''".
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| title = Su di un teorema di Hartogs
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  | mr = 0131663
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}}. A fundamental paper in the theory of Hrtogs' phenomenon. The typographical error in the title is reproduced in as it is appears in the original version of the paper.
*{{Citation
| last = Fichera
| first = Gaetano
| author-link = Gaetano Fichera
| title = Caratterizzazione della traccia, sulla frontiera di un campo, di una funzione analitica di più variabili complesse
| journal = Rendiconti della [[Accademia Nazionale dei Lincei]], Classe di Scienze Fisiche, Matematiche e Naturali
| series = series 8,
| volume = 22
| issue = 6
| pages = 706–715
| year = 1957
| language = Italian
| mr = 0093597
| zbl = 0106.05202
}}. An epoch-making paper in the theory of [[CR-function]]s, where the Dirichlet problem for [[Several complex variables|analytic functions of several complex variables]] is solved for general data. A translation of the title reads as:-"''Characterization of the trace, on the boundary of a domain, of an analytic function of several complex variables''".
*{{Citation
| last = Fichera
| first = Gaetano
| author-link = Gaetano Fichera
| title = Sul fenomeno di Hartogs per gli operatori lineari alle derivate parziali
| journal = Rendiconti dell' Istituto Lombardo di Scienze e Lettere. Scienze Matemàtiche e Applicazioni, Series A. 
| volume = 117
| pages = 199–211
| year = 1983
| language = Italian
| doi =
| mr = 0848259 
| zbl = 0603.35013
}}. An English translation of the title reads as:-"''Hartogs phenomenon for certain linear partial differential operators''".
*{{Citation
  | last = Fueter
  | first = Rudolf
  | author-link = Rudolf Fueter
  | title = Über einen Hartogs'schen Satz
  | journal = [[Commentarii Mathematici Helvetici]]
  | volume = 12
  | issue = 1
  | pages = 75–80
  | year = 1939–1940
  | language = German
  | url = http://retro.seals.ch/digbib/en/view?rid=comahe-001:1939-1940:12::10
  | doi = 10.5169/seals-12795
  | jfm = 65.0363.03 
  | mr = 
  | zbl = 0022.05802
}}. Available at the [http://retro.seals.ch/digbib/home SEALS Portal]. An English translation of the title reads as:-"''On a theorem of Hartogs''".
*{{Citation
  | last = Fueter
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  | author-link = Rudolf Fueter
  | title = Über einen Hartogs'schen Satz in der Theorie der analytischen Funktionen von <math>n</math> komplexen Variablen
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  | volume = 14 
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  | pages = 394–400
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  | doi = 10.5169/seals-14312
  | jfm = 68.0175.02
  | mr = 0007445
  | zbl = 0027.05703 
}} (see also {{Zbl|0060.24505}}, the cumulative review of several papers by E. Trost). Available at the [http://retro.seals.ch/digbib/home SEALS Portal]. An English translation of the title reads as:-"''On a theorem of Hartogs in the theory of analytic functions of <math>n</math> complex variables''".
*{{Citation
  | last = Hartogs
  | first = Fritz
  | author-link = Friedrich Hartogs
  | title = Einige Folgerungen aus der ''Cauchy''schen Integralformel bei Funktionen mehrerer Veränderlichen.
  | journal = Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften zu München, Mathematisch-Physikalische Klasse
  | language = German
  | volume = 36
  | pages = 223–242
  | year = 1906
  | url = 
  | doi = 
  | jfm = 37.0443.01
}}.
*{{Citation
  | last = Hartogs
  | first = Fritz
  | author-link = Friedrich Hartogs
  | title = Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselber durch Reihen welche nach Potentzen einer Veränderlichen fortschreiten
  | journal = [[Mathematische Annalen]]
  | language = German
  | volume = 62
  | pages = 1–88
  | year = 1906a
  | url = http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002260913
  | doi = 10.1007/BF01448415
  | jfm = 37.0444.01
}}. Available at the [http://www.digizeitschriften.de/ DigiZeitschriften].
*{{Citation
  | last = Hörmander
  | first = Lars 
  | author-link = Lars Hörmander
  | title = An Introduction to Complex Analysis in Several Variables
  | place = Amsterdam–London–New York–Tokyo
  | publisher = [[Elsevier|North-Holland]]
  | origyear = 1966
  | year = 1990
| series = North–Holland Mathematical Library
  | volume = 7
  | edition = 3rd (Revised)
  | url = 
  | doi = 
  | mr = 1045639 
  | zbl = 0685.32001
  | isbn = 0-444-88446-7
}}.
*{{Citation
  | last = Kaneko
  | first = Akira
  | author-link = Akira Kaneko
  | title = On continuation of regular solutions of partial differential equations with constant coefficients
  | journal = Proceedings of the Japan Academy
  | volume = 49
  | issue = 1
  | pages = 17–19
  | date = January 12, 1973
  | url = http://projecteuclid.org/euclid.pja/1195519488
  | doi = 10.3792/pja/1195519488
  | mr = 0412578
  | zbl = 0265.35008
}}, available at [http://projecteuclid.org/DPubS?Service=UI&version=1.0&verb=Display&handle=euclid Project Euclid].
*{{Citation
  | last = Martinelli
  | first = Enzo
  | author-link = Enzo Martinelli
  | title = Sopra una dimostrazione di R. Fueter per un teorema di Hartogs 
  | journal = [[Commentarii Mathematici Helvetici]]
  | volume = 15
  | issue = 1
  | pages = 340–349
  | year = 1942–1943
  | language = Italian
  | url = http://retro.seals.ch/digbib/en/view?rid=comahe-002:1942-1943:15::26
  | doi = 10.5169/seals-14896
  | mr = 0010729
  | zbl = 0028.15201
}}. Available at the [http://retro.seals.ch/digbib/home SEALS Portal]. An English translation of the title reads as:-"''On a proof by R. Fueter of a theorem of Hartogs''".
*{{Citation
  | last = Osgood
  | first = W. F.
  | author-link = William Fogg Osgood 
  | title = Lehrbuch der Funktionentheorie. II
  | place = Leipzig
  | publisher = [[Teubner Verlag|B. G. Teubner]]
  | series = Teubners Sammlung von Lehrbüchern auf dem Gebiet der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen 
  | volume = Bd. XX - 1
  | year = 1929
  | pages = VIII+307
  | language = German
  | edition = 2nd
  | url = https://books.google.co.jp/books?id=1pSzLtN4Qp4C&printsec=frontcover&hl=it&redir_esc=y#v=onepage&q&f=true
  | doi = 
  | jfm = 55.0171.02}}
*{{Citation
  | last = Severi
  | first = Francesco
  | author-link = Francesco Severi
  | title = Una proprietà fondamentale dei campi di olomorfismo di una funzione analitica di una variabile reale e di una variabile complessa 
  | journal = [[Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali]]
  | series = series 6,
  | volume = 15
  | pages = 487–490
  | year = 1932
  | language = Italian
  | jfm = 58.0352.05
  | zbl = 0004.40702
}}. An English translation of the title reads as:-"''A fundamental property of the domain of holomorphy of an analytic function of one real variable and one complex variable''".
*{{Citation
  | last = Severi
  | first = Francesco
  | author-link = Francesco Severi
  | title = A proposito d'un teorema di Hartogs
  | journal = [[Commentarii Mathematici Helvetici]]
  | volume = 15
  | issue = 1
  | pages = 350–352
  | year = 1942–1943
  | language = Italian
  | url =http://retro.seals.ch/digbib/en/view?rid=comahe-002:1942-1943:15::27
  | doi = 10.5169/seals-14897
  | mr = 0010730
  | zbl = 0028.15301
}}. Available at the [http://retro.seals.ch/digbib/home SEALS Portal]. An English translation of the title reads as:-"''About a theorem of Hartogs''".

==外部リンク==
*{{SpringerEOM|title=Hartogs theorem|last= Chirka|first= E. M. |urlname=Hartogs_theorem}}
*{{planetmath reference|id=10242|title=Failure of Hartogs' theorem in one dimension (counterexample)}}
*{{PlanetMath|urlname=HartogsTheorem|title=Hartogs' theorem}}
*{{planetmath reference|id=10238|title=Proof of Hartogs' theorem}}

{{DEFAULTSORT:はるとおくすのかくちようていり}}
[[Category:多変数複素函数論]]
[[Category:複素解析の定理]]
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